I also misread this initially.

I get 18 ways:

       set            product   power

[1, 2, 3, 4, 6, 9] 1296   = 6^ 4
[1, 2, 3, 6, 9, 24] 7776   = 6^ 5
[1, 2, 3, 6, 12, 18] 7776   = 6^ 5
[1, 2, 3, 8, 9, 18] 7776   = 6^ 5
[1, 2, 4, 6, 9, 18] 7776   = 6^ 5
[1, 2, 6, 9, 18, 24] 46656   = 6^ 6
[1, 3, 4, 6, 9, 12] 7776   = 6^ 5
[1, 3, 4, 9, 18, 24] 46656   = 6^ 6
[1, 3, 6, 9, 12, 24] 46656   = 6^ 6
[1, 3, 6, 9, 16, 18] 46656   = 6^ 6
[1, 3, 8, 9, 12, 18] 46656   = 6^ 6
[1, 4, 6, 9, 12, 18] 46656   = 6^ 6
[1, 6, 9, 12, 18, 24] 279936   = 6^ 7
[2, 3, 4, 9, 12, 18] 46656   = 6^ 6
[2, 3, 6, 8, 9, 18] 46656   = 6^ 6
[2, 3, 9, 12, 18, 24] 279936   = 6^ 7
[3, 4, 6, 9, 18, 24] 279936   = 6^ 7
[3, 6, 8, 9, 12, 18] 279936   = 6^ 7

------------

def product(aList):
    p = 1
    for i in aList:
        p *= i
    return p

def prime_factor(n):
    """  for integer n, return a list of all the prime factors """
    top = n // 2
    factors = []
    
    for i in range(2,top+1):
        while n/i % 1 == 0:
            factors.append(int(i))
            n = n/i
            if n == 1:
                return factors
    if n != 1:
        factors.append(int(n))
    return factors



import math
from itertools import combinations

maxProd = 25*24*23*22*21*20
minProd = 1*2*3*4*5*6
highestPower =  int( math.log(maxProd,6))
lowestPower =  int( math.log(minProd,6))+1

nums = [i for i in range(1,26)]
powers = [6**i for i in range(lowestPower, highestPower + 1)]

for comb in combinations(nums,6):
    if product(list(comb)) in powers:
        print(list(comb), product(list(comb)), '  = 6^',round(math.log(product(list(comb)), 6)))